The Curious Mathematics of Nothing – and Why It Is So Important.

Abstract: We usually think of mathematics as something kids have to learn in school – we rarely or only dimly comprehend that it provides the foundation for virtually all of modern science and the ubiquitous technological infrastructure that nurtures us. Mathematics is a marvel, and the greatest minds have not been able to explain why it works as well as it does in explaining the way the world works. Curiously, one of the more incomprehensible features of modern mathematics is – nothing.

The Curious Mathematics of Nothing – and Why It Is So Important.

Most of us live in a world dominated by human scientific and technological inventions. We may walk in the woods, toil in our gardens, visit natural wonders and pray to our God, but most of our time and attention is preoccupied by manufactured goods and devices: homes, cars, tools, appliances and (increasingly) computers and other digital devices. Moreover, we are largely ignorant of the complex processes by which these goods and devices are designed and manufactured, and of the scientific knowledge that underpins their functioning. Most of us are even further removed from the detailed empirical studies and mathematical formulae that comprise that knowledge. We usually think of mathematics as something kids have to learn in school (sometimes with great difficulty) – we rarely or only dimly comprehend that it provides the foundation for virtually all of modern science and the ubiquitous technological infrastructure that nurtures us.

Mathematics is a marvel, and the greatest minds have not been able to explain why it works as well as it does in explaining the way the world works. For example, there were more than 200 entries in the 2015 FQXi Essay contest, including my entry “The Hole at The Center of Creation”, each of which attempted, but failed, to crack that mystery.

Curiously, one of the more incomprehensible features of modern mathematics is – nothing. I address the issues involved in nothing (the void, zero, etc….) at some length in THTCC, but it is also the subject of several modern popular general interest books. Charles Siefe titled his 2010 Penguin book ZERO: The Biography of a Dangerous Idea. Amir Alexander recently tackled a related question in Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. (Xxxx books. 2015).

Siefe’s book explains why the concept of zero, or nothing, served as a “third rail” for the pursuit of clear, consistent, precise and definite mathematical proofs across the millennia, from Zeno to Newton to Cantor. Even the simple operations of subtraction and division, not to speak of more esoteric mathematical pursuits, founder on the properties of zero. For example, “1 x 0” is equal to “0” – not by proof but by definition, and “1 / 0” is undefined and therefore prohibited. Some of the most creative developments in higher mathematics have resulted from the effort to avoid, or find clever ways around, those properties.

Alexander takes this a step further, by claiming that one of those clever devices, a concept known as the infinitesimal, played a key role in the religious, economic and political upheavals of the late Middle Ages. The western world as we know it, and our liberal democratic institutions that so revere the freedom of human thought, would not exist but for the triumph of this mathematical work-around to the limitations of – nothing. This is an interesting thesis, and Alexander argues it well. But his proofs fall short on several grounds. Specifically, Alexander falls prey to one of the pitfalls of rational argument that we have outlined in our essays on Being Rational: His political and personal convictions bias his analysis and presentation – his mathematical reasoning skips a few beats, and his historical conclusions are speculative and probably wrong.

The two main historical periods of interest for Alexander are the Counter Reformation in the late 16th century, and the later 17th century interregnum period in England. The counter-reformation movement, led in large measure by the Jesuits, was an effort to rehabilitate the Catholic Church, so badly scarred by the reformation and the waves of religious liberalization that resulted. Among the successes of the Jesuits in beating back the protestant tide was the creation of a strong worldwide network of high quality schools and universities. Mathematics, particularly proof-based Aristotelian logic and Euclidean geometry became an integral core to the Jesuit pedagogy – and were particularly suited to a belief system that promoted the infallibility of the Pope. In this context, there was a counterpoint in the mathematical speculations of Cavalieri and other Italians (the concepts were supported by Galileo) regarding the notion of infinitesimals. This theory suggested that geometric proofs, resting on principles of abstract and dimension-less points, lines and planes were superfluous — the world itself was extended and all real objects have a spatial reality. Planes have thickness, lines have width and points take up space – you could find the area of a figure by adding up the width of all the lines that comprised it. Alexander claims that the victory of the Jesuits in wiping out the novel idea of infinitesimals in Italy set the country back several hundred years. Italy became a mathematical backwater and its earlier pre-eminence in the pursuit of mathematical and scientific discoveries was eventually lost.

In England, the battle for mathematical ideology played out between two figures: the well-known and extraordinarily pompous Thomas Hobbes, and the energetic John Wallis, an early member of the Royal Philosophical Society that included the greatest scientists and innovators of England. Hobbes came to Euclidean mathematics after publishing his popular philosophical and political tome, Leviathan, in which he proclaimed the necessity of a powerful central state to control individualist impulses that can ravage public order – as they had during the English Civil Wars when anarchistic protestants and monarchist Anglicans tore the country apart. He found geometric proof to be a perfect tool for his lifelong project of proving himself right, and he expanded it into the political and philosophical realm (for which, by the way, it is poorly suited). Wallis, in contrast, was allied with more liberal temperaments, and declared the primacy of induction and empirical inquiry over deductive proof. The infinitesimals of Cavalieri were perfectly matched to this perspective as they involved constructions (e.g. stacking an infinite number of parallel lines to create a particular shape) that could be studied experimentally rather than deductively. Wallis became an advocate of infinitesimals, and he and Hobbes began a lifelong and incredibly vicious feud spanning many decades. Hobbes’ viewpoint, and career, was eventually squelched, and the “new math” prospered, eventually giving rise to the calculus upon which modern world is built.

These are interesting theses, but as I pointed out earlier, they are not entirely convincing. For one thing, the historical trends Alexander reports were extraordinarily complex. It is quite a stretch in the history of ideas to claim that one idea – an obscure mathematical one at that – would have the potency to drive world events along the lines he suggests. It is rather more likely that the mathematical practice was a collateral factor that went along with but did not define the underlying historical trends.

The larger problem is that Alexander’s characterization of the mathematics of infinitesimals is incorrect – they are, in fact, seriously flawed for reasons that Siefe explores in details in his work. Cavalieri buried these flaws in his ponderous and largely impenetrable work – something any serious elite mathematician would recognize. Fermat, for example, as Alexander points out (page 274) was not impressed with Wallis’ work: “Wallis, Fermat argued, had it backwards: one cannot sum the lines of a figure unless one already knows the area of the figure, arrived at by traditional means.”

Moreover, no serious mathematician eschews deductive proof for inductive ones, as all inductive proofs are subject to the danger of falsification from a single counter-example. Every European believed that all swans were white – until black swans were discovered in Australia. In contrast, deductive mathematical proofs are self–evident and incontrovertible. From this vantage point, it may seem that Alexander has his thesis backwards. It may be that the Jesuits and later continental mathematicians (I am excluding Hobbes from this category) were actually more committed to the pursuit of truth that the so-called reformers, who seemed to find a justification for their less rigorous mathematics in the politics of the day.

The most disappointing aspect of Alexander’s argument is the lengthy focus on the personal feud between Wallis and Hobbes. Both were, if anything, second-rate mathematicians, and Hobbes’ philosophy and personality is deeply flawed. These two characters are not qualified for the task Alexander set out for them in his narrative, and his overall thesis suffers as a result. Nevertheless, his work is an incredibly entertaining and well-written account.

Admittedly, the curious controversies over infinitesimals, and other “nothings”, does not seem to provide much insight into the big questions of the day, be they philosophical, scientific or political. But it is worth noting that mathematics is, indeed, relevant to those questions, and has a bearing on history and politics. Indeed, the argument between physicalists (the physical world is all there is and math is a tool invented by humans) and platonists (there is a idealized abstract realm of mathematical concepts and structure apart from the physical) is as heated today in the FQXi essays as it was in the 15th or the 17th centuries.

There is much we can learn from an informed inquiry into the past. Significantly, it is important to remain humble about what we think we know.